# Variational Rectified Flow Matching

arXiv: https://arxiv.org/abs/2502.09616 · HTML: https://mlsanity.com/paper/2502.09616/

## Key facts

- Cited by 35 papers in the map (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 8 of its 33 references that are in the map.
- First posted on arXiv 13 Feb 2025.

## Placed in

- Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/)

## Builds on (most cited first, 8 of 8)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [SiT: Exploring Flow and Diffusion-based Generative Models with Scalable Interpolant Transformers](https://mlsanity.com/paper/2401.08740/) (2024)
- [Understanding Diffusion Models: A Unified Perspective](https://mlsanity.com/paper/2208.11970/) (2022)
- [DiffuseVAE: Efficient, Controllable and High-Fidelity Generation from Low-Dimensional Latents](https://mlsanity.com/paper/2201.00308/) (2022)
- [Consistency Flow Matching: Defining Straight Flows with Velocity Consistency](https://mlsanity.com/paper/2407.02398/) (2024)
- [PeRFlow: Piecewise Rectified Flow as Universal Plug-and-Play Accelerator](https://mlsanity.com/paper/2405.07510/) (2024)
- [Variational Flow Matching for Graph Generation](https://mlsanity.com/paper/2406.04843/) (2024)

## Built on by (most cited first, 30 of 35)

- [Towards Hierarchical Rectified Flow](https://mlsanity.com/paper/2502.17436/) (2025)
- [The Curse of Conditions: Analyzing and Improving Optimal Transport for Conditional Flow-Based Generation](https://mlsanity.com/paper/2503.10636/) (2025)
- [Adversarial Flow Models](https://mlsanity.com/paper/2511.22475/) (2025)
- [VFP: Variational Flow-Matching Policy for Multi-Modal Robot Manipulation](https://mlsanity.com/paper/2508.01622/) (2025)
- [Learning Straight Flows: Variational Flow Matching for Efficient Generation](https://mlsanity.com/paper/2511.17583/) (2025)
- [Hierarchical Rectified Flow Matching with Mini-Batch Couplings](https://mlsanity.com/paper/2507.13350/) (2025)
- [Efficient Flow Matching using Latent Variables](https://mlsanity.com/paper/2505.04486/) (2025)
- [Quantum Flow Matching](https://mlsanity.com/paper/2508.12413/) (2025)
- [Variational Masked Diffusion Models](https://mlsanity.com/paper/2510.23606/) (2025)
- [Training-Free Refinement of Flow Matching with Divergence-based Sampling](https://mlsanity.com/paper/2604.04646/) (2026)
- [A Minimum Variance Path Principle for Accurate and Stable Score-Based Density Ratio Estimation](https://mlsanity.com/paper/2602.00834/) (2026)
- [FlowSynth: Instrument Generation Through Distributional Flow Matching and Test-Time Search](https://mlsanity.com/paper/2510.21667/) (2025)
- [Temporal Pair Consistency for Variance-Reduced Flow Matching](https://mlsanity.com/paper/2602.04908/) (2026)
- [Variational Grey-Box Dynamics Matching](https://mlsanity.com/paper/2602.17477/) (2026)
- [CaReFlow: Cyclic Adaptive Rectified Flow for Multimodal Fusion](https://mlsanity.com/paper/2602.19140/) (2026)
- [Transition Flow Matching](https://mlsanity.com/paper/2603.15689/) (2026)
- [DiReCT: Disentangled Regularization of Contrastive Trajectories for Physics-Refined Video Generation](https://mlsanity.com/paper/2603.25931/) (2026)
- [Coupled Flow Matching](https://mlsanity.com/paper/2510.23015/) (2025)
- [FlowDet: Unifying Object Detection and Generative Transport Flows](https://mlsanity.com/paper/2512.16771/) (2025)
- [MixFlow: Mixed Source Distributions Improve Rectified Flows](https://mlsanity.com/paper/2604.09181/) (2026)
- [Second Order Drifting Models](https://mlsanity.com/paper/2608.07924/) (2026)
- [TracingFlow: A Simulation-Free Trajectory Inference Framework Based on Second-Order Dynamics](https://mlsanity.com/paper/2608.21070/) (2026)
- [Risk-Entropic Flow Matching](https://mlsanity.com/paper/2512.03078/) (2025)
- [Communication-Inspired Tokenization for Structured Image Representations](https://mlsanity.com/paper/2602.20731/) (2026)
- [Uncovering Physical Drivers of Dark Matter Halo Structures with Auxiliary-Variable-Guided Generative Models](https://mlsanity.com/paper/2602.23518/) (2026)
- [Towards Faster Language Model Inference Using Mixture-of-Experts Flow Matching](https://mlsanity.com/paper/2604.15009/) (2026)
- [Structured Coupling for Flow Matching](https://mlsanity.com/paper/2605.07676/) (2026)
- [Coreset-Induced Conditional Velocity Flow Matching](https://mlsanity.com/paper/2605.12951/) (2026)
- [The Reverse Telescoping Coordinate System for Positive Definite Matrices: Geometry, Computation, and Generative Modeling](https://mlsanity.com/paper/2606.15442/) (2026)
- [Field Codes for Distributed Coupling Samplers and Certified Empirical Transport](https://mlsanity.com/paper/2607.27078/) (2026)

## Abstract (arXiv)

We study Variational Rectified Flow Matching, a framework that enhances classic rectified flow matching by modeling multi-modal velocity vector-fields. At inference time, classic rectified flow matching 'moves' samples from a source distribution to the target distribution by solving an ordinary differential equation via integration along a velocity vector-field. At training time, the velocity vector-field is learnt by linearly interpolating between coupled samples one drawn from the source and one drawn from the target distribution randomly. This leads to ''ground-truth'' velocity vector-fields that point in different directions at the same location, i.e., the velocity vector-fields are multi-modal/ambiguous. However, since training uses a standard mean-squared-error loss, the learnt velocity vector-field averages ''ground-truth'' directions and isn't multi-modal. In contrast, variational rectified flow matching learns and samples from multi-modal flow directions. We show on synthetic data, MNIST, CIFAR-10, and ImageNet that variational rectified flow matching leads to compelling results.